Textbook

Calculus and Vectors Nelson
Chapter

Chapter 9
Section

Relations between Points, Lines and Planes Chapter Review

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Solutions
37 Videos

The lines `2x -y = 31`

,`x + 8y =-34`

, and

`3x+ky=38`

all pass through a common point.

Determine the value of `k`

.

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Q1

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) &x - y &=13\\
&(2) &3x + 2y &=-6\\
&(3) &x + 2y &=-19
\end{array}
```

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Q2

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) &x - y+ 2z &=3\\
&(2) &2x- 2y +3z &=1\\
&(3) &2x -2y +z &=11
\end{array}
```

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Q3a

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) &x + y+ z &=300\\
&(2) &x+ y -z &=98\\
&(3) &x -y +z &=100
\end{array}
```

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Q3b

a) Show that the points `(1, 2, 6), (7, -5, 1), (1, 1, 4)`

, and `(-3, 5, 6)`

all lie on the same plane.

b) Determine the distance from the origin to the plane you found in part a.

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Q4

Determine the following distances:

the distance from `A(-1, 1, 2)`

to the plane with equation `3x -4y - 12z - 8 = 0`

.

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Q5a

Find the distance from `B(3, 1, -2)`

to the plane with equation

```
\displaystyle
8x -8y + 4z - 7 =0
```

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Q5b

Determine the intersection of the plane `3x -4y -5z =0`

with `\vec{r} = (3, 1, 1) + t(2, -1, 2), t \in \mathbb{R}`

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Q6

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) &3x - 4y+ 5z &=9\\
&(2) &6x- 9y +10z &=9\\
&(3) &9x -12y +15z &=9
\end{array}
```

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Q7a

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) &2x + 3y+ 4z &=3\\
&(2) &4x+ 6y +8z &=4\\
&(3) &5x + y -z &=1
\end{array}
```

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Q7b

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) &4x -3y+ 2z &=2\\
&(2) &8x- 6y +4z &=4\\
&(3) &12x - 9y + 6z &=1
\end{array}
```

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Q7c

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) & 3x + 4y + z &=4 \\
&(2) & 5x + 2y+ 3z &=2\\
&(3) & 6x + 8y + 2z &=8
\end{array}
```

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Q8a

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) & 4x- 8y + 12z &=4 \\
&(2) & 2x + 4y+ 6z &=4\\
&(3) & x - 2y - 3z &=4
\end{array}
```

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Q8b

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) & x- 3y + 3z &=7 \\
&(2) & 2x - 6y+ 6z &=14\\
&(3) & -x + 3y - 3z &= -7
\end{array}
```

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Q8c

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) & 3x- 5y + 2z &=4 \\
&(2) & 6x + 2y -z &=2\\
&(3) & 6x - 3y + 8z &=6
\end{array}
```

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Q9a

Solve the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) & 2x- 5y + 3z &= 1 \\
&(2) & 4x + 2y+ 5z &=5\\
&(3) & 2x + 7y + 2z &=4
\end{array}
```

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Q9b

Determine the intersection of each set of planes, and show your answer geometrically.

```
\displaystyle
\begin{array}{llll}
&(1) & 2x + y + z &= 6 \\
&(2) & x - y - z &=-9\\
&(3) & 3x + y & = 2
\end{array}
```

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Q10a

Determine the intersection of each set of planes, and show your answer geometrically.

```
\displaystyle
\begin{array}{llll}
&(1) & 2x -y + 2z &= 2 \\
&(2) & 3x + y -z &=1\\
&(3) & x - 3y + 5z & = 4
\end{array}
```

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Q10b

Determine the intersection of each set of planes, and show your answer geometrically.

```
\displaystyle
\begin{array}{llll}
&(1) & 2x + y - z &= 0 \\
&(2) & x - 2y + 3z&=0\\
&(3) & 9x + 2y -z & =0
\end{array}
```

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Q10c

The line ` \vec{r} =(2, -1 -2) + s(1, 1, -2), x \in \mathbb{R}`

, intersects the xz-plane at point `P`

and the xy-plane `Q`

. Calculate the length of the line segment `PQ`

.

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Q11

a) Given the line `\vec{r} = (3, 1, -5) + s(2, 1, 0), x \in\mathbb{R}`

, and the plane `x - 2y + z + 4 = 0`

, verify that the line lies on the plane.

b) Determine the point of intersection between the line `\vec{r} = (7, 5, -1) + t(4, 3, 2), t \in \mathbb{R}`

, and the line given in part a.

c) Show that the point of intersection of the lines is a point on the lane given in part a.

d) Determine the Cartesian equation of the lane that contains the line`\vec{r} = (7, 5, -1) + t(4, 3, 2), t \in \mathbb{R}`

and is perpendicular to the plane given in part a.

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Q12a

a) Determine the distance from point `A(-2, 1, 1)`

to the line with equation `\vec{r} = (3, 0, -1) + t(1, ,1 2), t \in \mathbb{R}`

.

b) What are the coordinates of the point on the line that produces this shortest distance?

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Q13

You are given the lines `\vec{r} = (1, -1, 1)+ t(3, 2, 1), t\in \mathbb{R}`

, and `\vec{r} = (-2, -3, 0) + s(1, 2, 3), s \in \mathbb{R}`

.

a) Determine the coordinates of their point of intersection.

b) Determine a vector equation for the line that is perpendicular to both of the given lines and passes through their point of intersection.

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Q14

a) Determine the equation of the plane that contains

`L: \vec{r} = (1, 2, -3) +s(1, 2, -1), s\in \mathbb{R}`

and point `K(3, -2, 4)`

.

b) Determine the distance from point `S(1, 1, -1)`

to the plane you found in part a.

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Q15

Consider the following system of equations:

```
\displaystyle
\begin{array}{llll}
&(1) & x + y -z &= 1\\
&(2) & 2x - 5y + z &=-1\\
&(3) & 7x - 7y - z & = k
\end{array}
```

a) Determine the value(s) of `k`

for which the solution to this system is a line.

b) Determine the vector equation of the line.

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Q16

Determine the solution to each system of equations.

```
\displaystyle
\begin{array}{llll}
&(1) & x + 2y + z &= 1\\
&(2) & 2x - 3y -z &=6\\
&(3) & 3x + 5y + 4z & =5\\
&(4) & 4x + y + z & = 8
\end{array}
```

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Q17a

Determine the solution to each system of equations.

```
\displaystyle
\begin{array}{llll}
&(1) & x - 2y + z &= 1\\
&(2) & 2x - 5y +z &=-1\\
&(3) & 3x - 7y + 2z & =0\\
&(4) & 6x - 14y + 4z & = 0
\end{array}
```

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Q17b

Solve the following system of equations for `a, b`

, and `c`

:

```
\displaystyle
\begin{array}{llll}
&(1) & \frac{9a}{b} - 8b +\frac{3c}{b} &= 4\\
&(2) & \frac{-3a}{b} +4b + \frac{4c}{b} &= 3\\
&(3) & \frac{3a}{b} + 4b - \frac{4c}{b} & =3
\end{array}
```

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Q18

Determine the point of intersection of the line `\frac{x + 1}{-4} = \frac{y - 2}{3} = \frac{z - 1}{-2}`

and the plane with equation `x + 2y -3z + 10 =0`

.

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Q19

Point `A(1, 0, 4)`

is reflected in the plane with equation `x -y + z - 1 = 0`

. Determine the coordinates of the image point.

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Q20

The three planes with equations `3x + y + 7z + 3 = 0`

, `4x -12y + 4z -24 = 0`

, and `x + 2y +3z - 4= 0`

do not simultaneously intersect.

Considering the planes in pairs, determine the three lines of intersection.

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Q21a

The three planes with equations `3x + y + 7z + 3 = 0`

, `4x -12y + 4z -24 = 0`

, and `x + 2y +3z - 4= 0`

do not simultaneously intersect.

Show that these three lines are parallel.

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Q21b

Determine the equation of a parabola that has its axis parallel to the y-axis and passes through the points `(-1, 2), (1, -1)`

, and `(2, 1)`

. (Note that the general form of the parabola that is parallel to the y-axis is `y = ax^2 + bx + c`

.)

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Q23

A perpendicular line is drawn from point `X(3, 2, -5)`

to the plane `4x - 5y + z -9=0`

and meets the plane at point `M`

. Determine the coordinates of M.

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6.29mins

Q24

Determine the values of `A, B`

, and `C`

if the following is true:

```
\displaystyle
\frac{11x^2 -14x +9}{(3x -1)(x^2 + 1)} = \frac{A}{3x -1} + \frac{Bx + C}{x^2 + 1}
```

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3.51mins

Q25

A line L is drawn through point D, perpendicular to the line segment EF, and meets EF at point J.

a. Determine an equation for the line containing the line segment EF.

b. Determine the coordinates of point J on EF.

c. Determine the area of `\triangle DEF`

.

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Q26

Determine the equation of the plane that passes through `(5, -5, 5)`

and is perpendicular to the line of intersection of the planes `3x - 2z + 1 = 0`

and `4x + 3y + 7 = 0`

.

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Q27